Physics Letters B 788 (2019) 535–541

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Physics Letters B

www.elsevier.com/locate/physletb

Constraining P and CP violation in the main decay of the neutral Sigma ∗ Shankar Sunil Nair, Elisabetta Perotti, Stefan Leupold

Institutionen för fysik och astronomi, Uppsala Universitet, Box 516, S-75120 Uppsala, Sweden a r t i c l e i n f o a b s t r a c t

Article history: On general grounds based on quantum field theory the decay amplitude for 0 → γ consists of a Received 13 February 2018 conserving magnetic and a parity violating electric dipole transition moment. Because of the subsequent Received in revised form 30 August 2018 self-analyzing weak decay of the hyperon the interference between magnetic and electric dipole tran- Accepted 9 September 2018 sition moment leads to an asymmetry in the angular distribution. Comparing the decay distributions for Available online 28 November 2018 the 0 hyperon and its antiparticle gives access to possible C and CP violation. Based on flavor SU(3) Editor: V. Metag symmetry the present upper limit on the can be translated to an upper Keywords: limit for the angular asymmetry. It turns out to be far below any experimental resolution that one can Radiative decays of expect in the foreseeable future. Thus any true observation of a CP violating angular asymmetry would Electric dipole moment constitute physics beyond the , even if extended by a CP violating QCD theta-vacuum- CP violation angle term. © 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction cle decay, 0 → γ , and for the corresponding antiparticle decay, ¯ 0 → γ ¯ , one can search for C and CP violation. There is much more matter than antimatter in the universe. If Let us add right away that the conservative expectation is that this is not just by chance but has a dynamical origin, then an ex- one would not find an effect of P, C or CP violation in this decay. planation of this asymmetry should come from the realm This will be substantiated by our explicit estimates given below. of . Based on Sakharov’s conditions [1]this has Yet, if theory predicts that something is very small, then it might spurred the search for baryon decays and reactions that show CP be worth checking this experimentally. Even if one “only” estab- violation [2]. (Here C denotes charge conjugation symmetry and lishes an upper limit, this can help to constrain beyond-standard- Pparity symmetry.) Two directions of such searches are weak de- model developments. Needless to add that if one found an effect cays of [3–5] and electric dipole moments (EDMs) [6,7]. not predicted by established theory, then this would be sensa- In the present work we propose to study a reaction that is in tional. between these two types of reactions, the decay of the neutral Actually already in 1962 the interplay of magnetic and electric ground-state Sigma hyperon to a photon and a Lambda hyperon, dipole transition moment for the 0 decay has been addressed in 0 → . This electromagnetic baryon decay could in principle γ [9], though under the implicit assumption of CP conservation. In show an interference between a parity conserving and a parity vi- the present work we will be more general. In addition, to the best olating amplitude. The latter would come from an electric dipole of our knowledge, the search for such interference effects, albeit transition moment (the former from a magnetic transition mo- suggested so early, has never been conducted. With recent and on- ment). Note that the hyperon decays further into pion and going experiments on hyperon production (e.g. at ELSA [10], J-LAB proton on account of the [8]. The possible in- [11,12], GSI [13,14], BEPC II [15,16], KEKB [17], CESR [18]) and terference in the first decay can then be observed as an angular in particular with the upcoming PANDA experiment [19,20]as a asymmetry in the decay products of the, in total, three-body decay − hyperon–antihyperon factory, it is absolutely timely to establish at 0 → γπ p. Comparing the asymmetry parameters for the parti- least upper limits for the angular asymmetry of the 0 decay. The process 0 → γ constitutes the main decay of the 0 * Corresponding author. ground state . Its lifetime is governed by this decay [8]. There- E-mail address: [email protected] (S. Leupold). fore the 0 lives much longer than hadronic resonances that decay https://doi.org/10.1016/j.physletb.2018.09.065 0370-2693/© 2018 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 536 S.S. Nair et al. / Physics Letters B 788 (2019) 535–541 on account of the , but much shorter than weakly 0 decay. Finally a summary and an outlook will be provided in decaying particles. This makes the analysis of 0 decays challeng- section 5. ing, as already pointed out in [9]. Hopefully these problems can be diminished by the increasing production rate of 0 particles and 2. Transition moments and angular decay asymmetries detector quality. 0 Let us relate and compare the process → γ in more detail For the coupling of baryons to the electromagnetic current Jμ, to the mentioned searches for CP violation in weak baryon de- we follow in principle [28]but adopt the definition of the photon cays and for EDMs. In our case there is an interference between momentum to our decay process: two amplitudes, a large and a small one. The first one is related    μ ¯  μ to the magnetic transition moment. This amplitude is parity con- B (p )| J |B(p)=e uB (p )  (q) uB (p) (1) serving and compatible with Quantum Electrodynamics (QED) and  with q := p − p and (QCD). The second one is related to   the electric dipole transition moment and is parity violating. Thus m  − m μ = μ + B B μ 2 one expects small effects right away. This is in contrast to similar  (q) γ q F1(q ) q2 radiative decays where both amplitudes are caused by the weak   0 → μ 2 μ 2 interaction, for instance  γ [21]; see also the note on “Ra- + i γ q + (mB + mB ) q γ5 F A (q ) diative Hyperon Decays” from the Particle Data Group [8]. For the weak radiative decays the angular asymmetry is appreciably large, i μν 2 − σ qν F2(q ) but the difference between particle and antiparticle decay is in mB + mB most cases so far unmeasurably small. Only recently first evidence 1 μν 2 − σ qνγ5 F3(q ). (2) for a CP violating decay of the b baryon has been reported [5]. mB + mB For our case we expect very small effects for the angular asymme-  try and for the particle–antiparticle differences. If B and B have the same intrinsic parity then the functions 2 2 Though nature violates all discrete symmetries P, C and CP,1 this F1(q ) and F2(q ) are the P conserving Dirac and Pauli transition 2 2 happens to different degrees and different interactions behave dif- form factors. F A (q ) and F3(q ) are the P violating Lorentz invari- ferently. Therefore in an analysis of the decay 0 → γ it is useful ant transition form factors and are termed the anapole form factor to distinguish conceptually between a scenario of C violation but and the electric dipole form factor, respectively. We note in pass- 2 CP conservation and a scenario of C conservation but CP violation. ing that ideas how to access the q dependence of F1 and F2 for 0 We will present observables to test both scenarios for the process the transition of to have been presented in [30]. of interest. Yet for our concrete estimates we will focus on strong For the neutron, the Pauli form factor at the photon point is CP violation. related to the anomalous magnetic moment by [8] From an experimental point of view, the strong interaction con- = ≈− serves P, C and CP separately. Yet from the theory side it would F2,n(0) κn 1.91 (3) be natural to expect CP violation in QCD based on the non-trivial while the EDM of the neutron is given by topological structure of the non-abelian [23]. Such a e “theta-vacuum-angle term” is C conserving but P and CP violat- dn = F3,n(0). (4) ing. In particular, it gives rise to an EDM of the neutron. So far, no 2mn EDM of the neutron has been experimentally established [24], but The decay 0 → γ is only sensitive to the magnetic (dipole) the very small upper limit raises the question of why CP violation transition moment [8,30] in the strong sector is so unnaturally small (the “strong-CP prob- lem”) [25]. Note that the weak CP violation leads to an EDM of the κM := F2,(0) ≈ 1.98 (5) neutron that is many orders of magnitude below the experimental upper limit [6]. Therefore we focus on possible strong CP violation and the electric dipole transition moment (EDTM) in the following. As we will show below, our decay 0 → γ is e d := F (0). (6) related to the neutron EDM via the approximate SU(3) flavor sym- + 3, m0 m metry [26]. In the framework of chiral perturbation theory, one = can take care of the explicit symmetry breaking in a systematic The (transition) charge must vanish, F1,(0) 0, and the anapole way [27–29]. Based on the experimental upper limit of the neu- moment F A, does not contribute for real photons, technically 2 μ tron EDM we will provide an upper limit for the CP violating effect based on q = 0 and q μ = 0 where μ denotes the polarization on the Sigma decay chain. vector of the photon. In section 4 we will relate the electromag- 0 The rest of the paper is structured in the following way: In netic properties of the neutron and of the – transition. − the next section we will present a general parametrization of the Following [8]the successive decay → π p is parametrized transition moments for radiative decay amplitudes of baryons, dis- by the matrix element cuss the impact of C and CP symmetry/violation and relate all this M = ¯ A − B to the angular asymmetries of baryon and antibaryon decays. In 2 u p( γ5) u (7) section 3 we will calculate all relevant decay widths and angu- where A and B are complex numbers. To stay in close analogy lar distributions. In section 4 we will determine an upper limit for to this parametrization we write the decay matrix element for the those effects caused by strong CP violation, which is in turn related first decay 0 → γ as to the QCD theta-vacuum-angle term. Here baryon chiral perturba- tion theory [27–29]can be used to relate the neutron EDM to the M = ¯ − − ν μ∗ 1 u (a σμν b σμνγ5) u0 ( i)q . (8) The two decay parameters a and b are related to the transition 1 In quantum field theory the third discrete symmetry, time reversal symmetry T, moments via is intimately tied to CP via the CPT theorem [22]. In our decay, time reversal ar- guments are not directly applicable because we do not study the inverse formation e a = κ , b = id . (9) process γ → 0. Therefore we focus on CP instead of T. + M m0 m S.S. Nair et al. / Physics Letters B 788 (2019) 535–541 537

The decay asymmetries that will finally show up in the angular and − distribution of the three-body decay 0 → γπ p are defined by 2r sin δ = = [ − ± ◦] (see, e.g., [8]) tan φ tan ( 6.5 3.5) (18) 1 − r2 ∗ 2Re(a b) This yields (we do not go through the exercise of carrying along 0 := (10) α 2 2 |a| +|b| the (correlated) errors of α and φ) and ≈ ≈− ∗ r 0.37 ,δ 0.13 . (19) 2Re(s p) α := , (11) ¯ ¯ |s|2 +|p|2 For the antiparticle amplitudes s and p one must obtain the same ratio r as for the particle case: r =|p¯ /s¯|. But the relative with s := A and p := ηB. We have introduced η := |p p|/(mp + E p) ¯ ¯ phase δ¯ between s and p will be different from δ. The decays − + to compensate for the p-wave phase space relative to the s-wave. → π p and ¯ → π p¯ are governed by the weak interaction  Here mp denotes the mass of the proton and E p (p p ) its energy [31]. Here the CP violation is very small. The pion–baryon sys- (three-momentum) in the rest frame of the decaying into pion tem shows a strong final-state interaction. Therefore the phase and proton. between the amplitudes s¯ and p¯ is not small. If one ignores CP To reveal C and/or CP violation for our processes one has to violation one obtains δ = δ ¯ + π , which leads to α =−α ¯ . So compare particle and antiparticle decays. To this end we introduce far the experimental results are compatible with this assumption the following matrix elements that correspond to (7) and (8): of (approximate) CP conservation. According to [8]the current ex- M =−¯ A¯ − B¯ perimental status is 2¯ v( γ5) v p , (12) M =−¯ ¯ − ¯ − ν μ∗ =− ± 1¯ v0 (a σμν b σμνγ5) v ( i)q . (13) α¯ 0.71 0.08 , (20) In analogy to (10) and (11)we also introduce the asymmetries which should be compared with (17). ¯∗ ¯ := 2Re(a b) 3. Integrated and differential decay widths and angular α¯ 0 (14) |a¯|2 +|b¯|2 distributions and ∗ The matrix element (8)gives rise to the decay width 2Re(s¯ p¯ ) α ¯ := , (15) 2 2 3 |¯|2 +|¯ |2 (m − m ) s p = | |2 +| |2 0 0→γ ( a b ) . (21) ¯ ¯ 8πm3 with s¯ := A and p¯ := ηB. 0 C and CP conservation/violation influence the phases of the pa- For later use we also provide the partial decay width of the rameters a and b. In general, the products of a decay show some hyperon: final-state interaction that leads to an additional phase; see, e.g., 1/2 2 2 2 [2] and the note on “Baryon Decay Parameters” in [8]. It is useful to λ (m,mp,mπ )  → − = R (22) distinguish these two effects. To simplify things we note, however, π p 3 16πm that overall phases can be freely chosen in quantum mechanics. Only phase differences matter. Consequently we decompose with the Källén function 2 2 2 iδF λ(a, b, c) := a + b + c − 2(ab + bc + ac) (23) b = bW e , a = aW (16) with the parameters from the direct decay without final-state in- and [32] teractions labeled by an index W . The phase difference δ between F := |A|2 + 2 − 2 +|B|2 − 2 − 2 a and b encodes the final-state interaction. For this final-state in- R ((m mp) mπ ) ((m mp) mπ ). (24) teraction between and γ we assume that it is dictated by (P and In general, the corresponding decay widths for the antiparticle de- C conserving) QED, which implies that it is the same for –γ and A B ¯ cays are obtained by replacing the parameters a, b, , by the –γ . In Appendix A we demonstrate that the parity conserving corresponding parameters for the antiparticles. − contribution a to the decay amplitude can be chosen real and pos- For the full three-body decay 0 → γ pπ , shown on the left- itive for the particle and the antiparticle decay. CP conservation hand side of Fig. 1, we start with the double-differential decay would imply that the parity violating contribution b is real, while W width [8] C conservation would imply that bW is purely imaginary; see Ap- 2   pendix A for details. d  0→ − 1 1 γ pπ = |M |2 Since for our parameter α 0 the relative size between a and b 3 , (25) dm2 dm2 (2π)3 32m3 will matter, it might be illuminating to provide the corresponding 23 12 0 information for α and its building blocks s and p (or A and B) 2 := + 2 2 := + where we have introduced m23 (pγ p p ) and m12 (p p 2 and also for the antiparticle amplitudes. The overall strength of pπ ) . Our decay proceeds via the very narrow hyperon as an A and B enters the decay width of the , see (22) and (24)be- 2 intermediate state. Consequently we will integrate over m12 and low. Here we concentrate on the relative strength and present the focus on a single-differential decay width. In addition, we rewrite ratio r := |p/s| and the relative phase δ between s and p, i.e. 2 the m23 dependence into a dependence on the angle θ between =: iδ p/s re . The Particle Data Group provides the following infor- proton and photon defined in the rest frame of . Note that in this mation [8]:  =   =− frame we have p0 pγ and p p pπ which defines a plane. 2r cos δ This plane is depicted on the right-hand side of Fig. 1. One finds = α = 0.642 ± 0.013 (17) for the single-differential decay rate: 1 + r2 538 S.S. Nair et al. / Physics Letters B 788 (2019) 535–541

− := − with the branching ratio Br→π p →π p/ and the asym- metries defined in (10), (11). In terms of the number of events, N, this reads dN N = (1 − αα 0 cos θ). (32) d cos θ 2 For the corresponding antiparticle decay chain one finds

dN¯ N¯ = (1 − α ¯ α ¯ 0 cos θ), (33) d cos θ 2 0 → − Fig. 1. Left-hand side: Feynman diagram for the decay process γ pπ . Right- with the number of events N¯ . hand side: The decay process as seen in the rest frame. If one found an angular asymmetry in the particle and the an-

tiparticle decays, then one could determine the parameters α 0 d0→γ pπ − and α ¯ 0 since the asymmetries α and α ¯ have been measured d cos θ  [8]. According to the definitions (10), (14) and the relations given   2 − 2 1 1 m 0 m = 2 |M |2 12 in the appendix an observable to test CP symmetry is given by dm12 3 (2π)3 64m3 m2 0 12 OCP := α 0 + α ¯ 0 , (34) × 1/2 2 2 2 λ (m12,mp,mπ ). (26) i.e. this quantity vanishes if CP is conserved. Thus a non-vanishing M In (25), (26)the matrix element 3 for the three-body decay value signals CP violation. A corresponding observable to test is given by C symmetry is

M = u¯ (A − Bγ )(p/ + m )(a σ − b σ γ ) O := − 3 p 5 μν μν 5 C α0 α¯ 0 . (35) × − ν μ∗ 2 u0 ( i)pγ D(m12) Before we move on to an estimate of these parameters, it 2 should be stressed that a measurement of the single-differential =: Mred D(m ) (27) 12 decay rate provides only the product of the two asymmetry pa- with a resonance propagator rameters for the Sigma and the Lambda decay. To understand the implications of a non-vanishing experimental result for this prod- := − 2 + −1 D(s) (s m im) (28) uct, we discuss first the particle case alone and then the com- parison of particle and antiparticle case. The present situation is and the total width  = 1/τ where τ denotes the lifetime of that the asymmetry parameter of has been determined and is the . non-vanishing, see (17), while the asymmetry parameter of the Using the fact that the lifetime of the is extremely long [8] 0 has never been measured and is probably extremely small (see one can carry out the integral that appears in (26): next section). An experimental finding of a non-vanishing value for    2 − 2 the product would therefore be extremely interesting, irrespective m 0 m12 dm2 |M |2 λ1/2(m2 ,m2 ,m2 ) of the result for the antiparticle case. If this product is different 12 3 2 12 p π m12 for particle and antiparticle decays one should, of course, ana-    2 − 2 lyze whether the origin lies in the first or the second decay or m 0 m12 ¯ = dm2 |D (m2 )|2 |M |2 λ1/2(m2 ,m2 ,m2 ) in both. Since the asymmetry parameters for and can also 12 12 red 2 12 p π m be obtained from other production/decay chains [8], one needs a  12   1 m combined analysis of all these decays. But at present the situation = dm2 |M |2 12 2 2 red is such that the Lambda decays are compatible with the absence m (m − m )2 + (m  )2 12 (smallness) of CP violation. Therefore the observation of CP viola- m2 − m2 tion via non-trivial results for the angular distributions (32) and × 0 12 1/2 2 2 2 λ (m12,mp,mπ ) (33)would be sensational, no matter which of the successive de- m2 12 cays is influenced by CP violation.   2 − 2 π m 0 m ≈ |M |2 λ1/2(m2 ,m2 ,m2 ) (29) red 2 p π 4. Parameter estimates m m where we have used We will now use three-flavor chiral perturbation theory in the m  presence of a theta vacuum angle [27–29]to relate the neutron ≈ 2 − 2 0 πδ(m12 m). (30) EDM to the EDTM for the transition of the to the . The ex- m2 − m2 2 + m 2 ( 12 ) ( ) perimental upper limit for the former will provide an upper limit This replacement is completely appropriate when comparing to the for the latter and in this way an upper limit for the decay asym- experimental procedure where the intermediate is actually mea- metry based on the possible strong CP violation. sured by a displaced vertex. EDMs for nucleons and hyperons have been calculated in M [27–29]but the EDTM for the Sigma-to-Lambda transition has not Note that the reduced matrix element red is now evaluated 2 been addressed. In the present work we provide the result for this for onshell momenta of the . Then the average |Mred| depends EDTM from a complete calculation at the first non-trivial order in only on m2 or, in other words, on the angle θ between proton and 23 the chiral expansion. This calculation encompasses tree-level con- photon. Finally one obtains for the single-differential decay rate: tributions from the next-to-leading-order (NLO) baryon Lagrangian − and one-loop diagrams with one vertex from the NLO baryon La- d0→γ pπ 1 = − −  0→ Br→π p (1 αα 0 cos θ) (31) grangian but the rest from the leading-order (LO) baryon and LO d cos θ 2 γ S.S. Nair et al. / Physics Letters B 788 (2019) 535–541 539

¯ Table 1 in powers of the field θ. These functions appear as coefficients in Coefficients for the loop contributions to the EDTM of 0 to . The first column the chiral expansion of the meson Lagrangian [33,34]. For technical shows all possible meson–baryon pairs that can contribute in the loop. The second details we refer to [28,29]. and third columns contain the respective coefficients Cce and Cdf. The indices c–f The one-loop diagrams that contribute at the same (first non- refer to the loop diagrams depicted in [29]. The parameters F (bF ) and D (bD ) are low-energy constants from the LO (NLO) baryon Lagrangian. trivial) order as the tree-level terms (36) and (37)yield the follow- ing results: loops Cce Cdf { + −}− π , 4DbF 4FbD ¯ (2) − + 4eθ V {π , }−4DbF 4FbD loop 0 0 + − d =− √ (Cce − Cdf) J MM(0) (39) {K ,  }−(3F − D)(bD + bF )(D + F )(3bF − bD ) 4 − 3Fπ { } {K , p}−(D + 3F )(bD − bF )(D − F )(3bF + bD ) M,B and [29]

Table 2 (2) 8eθ¯ V Coefficients for the loop contributions to neutron loop =− 0 0 dn Ccd J MM(0). (40) EDM. See the caption of Table 1 for further details. F 4 π {M,B} loops Ccd − {π , p} 2(D + F )(bD + bF ) The sum covers the meson–baryon pairs listed in Table 1 or 2, + − {K , }−2(D − F )(b − b ) D F respectively. The loop function J MM is essentially the bubble dia- gram with two meson lines of mass M.3 It is given by  meson Lagrangians. Since the mass splitting between the members d 2 d k 1 of the baryon octet emerges only from the NLO baryon Lagrangian, J MM(q ) = i (2π)d (k2 − M2 + i )((k + q)2 − M2 + i ) it is a consistent procedure to neglect the mass difference between   (41) 0 and for the loop calculation. In practice, this drastically sim- 1 M2 σ − 1 2 = 2L + ln − 1 − σ ln , plifies the loop calculation. 16π 2 μ2 σ + 1 Since we follow exactly the framework of [29], we refer to this  reference for further details. In the following we will need the where σ = 1 − 4M2/q2 and L contains the divergence for space– expressions for the neutron EDM determined in [29] and for the time dimension d = 4as will be shown below. 0 – EDTM calculated here. Loops with various meson–baryon Following still [29]the loop divergence for the neutron case can combinations contribute. They are collected in Table 1 for the  be absorbed into the renormalization of w13: 0– transition and in Table 2 for the neutron. Note that the tree- level part shows the SU(3) flavor symmetry while the loops break (2)   r 24V 0 it due to the different meson masses that appear already at LO in w = w (μ) + (DbF + FbD )L (42) 13 13 F 4 the mesonic chiral Lagrangian. The important point is that chiral π perturbation theory provides a systematic expansion around the  r  where w13(μ) is the finite part of w13. The dependence on the flavor symmetric chiral limit where the masses vanish and renormalization scale μ comes through the quantity the baryon masses of the members of a multiplet are degenerate.   d−4 The tree-level results are given by μ 1 1  L = − [ln(4π) +  (1) + 1] (43) 4 (4π)2 d − 4 2 tree =−√ ¯ +  d eθ0 (αw13 w13) (36) 3 and cancels with that of the loops, leaving the EDM scale inde- and [29] pendent. It is a non-trivial check of our results (36) and (39) that the very same combination of loop divergence and counterterm tree 8 ¯  appears. d = eθ0 (αw13 + w ), (37) n 3 13 Finally, we spell out the numerical values of the low-energy  constants that we use for our size estimate of the (upper limit of with low-energy constants w13 and w from the baryon NLO La- 13 0 = := (2) (1) 2 the) – EDTM: To a large extent we follow [29] and use D grangian and α 144V 0 V 3 /(F0 Fπ Mη0 ) . Here, Fπ denotes the −1 −1 0.804, F = 0.463, bD = 0.068 GeV , bF =−0.209 GeV , Fπ = pion decay constant while F0 (Mη ) is the decay constant (mass) (2) − 0 92.2MeV, and V =−5 × 10 4 GeV4. of the singlet eta field. This field is intimately tied to the field 0  The numerical values of the low-energy constants w r (μ) and θ(x) that encodes the strong CP violation. In the framework of 13 w are not separately known. However, there is a recent lattice- chiral perturbation theory it is useful to introduce a chirally invari- 13 QCD determination of the counterterm combination [35] ant combination θ(¯ x) of these two fields. The vacuum expectation value of θ(¯ x) can be linked back to the vacuum expectation value := +  r wa(μ) αw13 w13(μ), of the original θ(x) field via (44) = ± −1 wa(1GeV) (0.04 0.01) GeV , −1 (2) 2 − 2 ¯ 4V 0 4M K Mπ which is exactly the quantity entering the tree-level expressions θ0 = 1 + θ0 (38) F 2 2 2 − 2 (36) and (37). π Mπ (2M K Mπ ) With these ingredients we obtain with the masses of pion, Mπ , and kaon, M K . Finally, the quantities ( j) V are the Taylor coefficients of the functions V ¯ expanded tree loop i i(θ) d d + d = ≈−0.88 . (45) d tree + loop n dn dn 2 Note that the more complicated loop structures caused by the different masses of the two external baryons of our transition do not appear at all for the “direct” EDMs addressed in [27–29]where initial and final states have always the same 3 We have here a slight misuse of M denoting both the meson mass and the mass. meson itself. This should not cause any ambiguities. 540 S.S. Nair et al. / Physics Letters B 788 (2019) 535–541

Amusingly, this is numerically close √to the ratio obtained just from In principle, also P violation without or with little CP violation tree tree =− ≈− the tree-level parts: d /dn 3/2 0.87. Of course, one can lead to an angular asymmetry. This can be caused for instance knows this only after performing the loop calculation. by the weak theory. An estimate of this effect is beyond the scope The experimental upper limit for the neutron EDM can be of the present paper. Yet we would like to stress that the CP viola- translated to an upper limit for the 0– EDTM: tion in the weak theory is very small. Thus an observation beyond the tiny effect (51)of a CP violating angular asymmetry as de- −26 duced from the 0 and ¯ 0 decays would point to physics beyond |d|≤2.5 × 10 e cm (46) the standard model. where we have utilized the current experimental upper bound of Our concrete relation between the EDM of the neutron and the | exp| ≤ × −26 0 → the neutron EDM [36], dn 2.9 10 e cm. EDTM of is based on chiral perturbation theory where Now we have all ingredients for an estimate of the asymmetry the effects from possible strong CP violation can be systematically included. The starting point for establishing this relation is the ap- α0 defined in (10). Using (9)together with (16) and anticipating 0 |a| |b| we obtain proximate SU(3) flavor symmetry that assigns the neutron, the and the to the very same multiplet. Chiral perturbation theory 2d sin δF takes care of the explicit breaking of this approximate symmetry in α 0 ≈− . (47) a a systematic way. But the approximate SU(3) flavor symmetry is a physical effect, not restricted to the standard model extended by a We utilize the first relation of (9) and the experimental value of theta-vacuum-angle term. Thus, in principle, it should be possible the magnetic transition moment (5). This gives us the upper limit to relate the EDM of the neutron and the EDTM of 0 → also in other beyond-standard-model frameworks that propose apprecia- | |≤ × −12| | α0 3.0 10 sin δF . (48) bly large EDMs. Yet, in practice, a quantitative relation between the two quantities requires an effective field theory that incorporates For the most conservative limit one might use for the phase caused (a) the electroweak theory, (b) additional terms that encode the | | ≤ by final-state interactions the mathematical bound sin δF 1. beyond-standard-model physics, and (c) chiral perturbation theory However, given that the final-state interaction between photon to account for the flavor symmetry breaking on the hadron level. and is an electromagnetic effect, one would rather estimate At present, not even a marriage of the electroweak theory and chi- | sin δF | ∼ αQED with the fine-structure constant αQED. For the elec- ral perturbation theory has been fully achieved. Consequently, the trically neutral the interaction with the photon is dominantly task to establish a completely model-independent relation between magnetic and therefore further kinematically suppressed by the the EDM of the neutron and the EDTM of 0 → remains as a fu- photon momentum. Therefore it feels justified to assume | sin δF | ≤ ture endeavor. − 10 2, which leads to Acknowledgements −14 |α 0 |≤3.0 × 10 . (49) We thank the reviewer for pointing out the importance of loop For the slope of the angular asymmetry in (32)we deduce from diagrams for our calculation. (49)an upper limit of Appendix A. Phases and discrete symmetries | |≤ × −14 αα0 1.9 10 (50) An effective Lagrangian for the 0– transition takes the form where the experimental value of the decay asymmetry for → − 1 1 p is given in (17). L = ¯ 0 μν + ¯ ¯ 0 μν π 0 aW σμν F aW σμν F Finally we turn to the antiparticle decay chain. We obtain here 2 2 1 1 the very same estimate as in (47), because the QCD theta-vacuum- ¯ 0 μν ¯ ¯ 0 μν − bW σμνγ5 F − bW σμνγ5 F . angle term conserves C and violates CP; see also the discussion in 2 2 the appendix. Thus it is useful to provide a corresponding upper (A.1) limit for the CP-test observable (34): ¯ =− ∗ ¯ = ∗ Hermiticity requires bW bW and aW aW . We still have the freedom to redefine the field 0 by an arbitrary phase. We choose |O |≤ × −14 CP 6.0 10 . (51) this phase such that aW = a¯ W is positive and real. Using the transformation properties of bilinears [37]it 5. Summary and outlook is easy to show that the a terms in our interaction Lagrangian (A.1) conserve P and C symmetry while the b terms break P. In addition, = ¯ We have provided a framework to search for P and CP violation C symmetry implies bW bW (i.e. bW is purely imaginary). Note in the decay 0 → γ utilizing the subsequent weak decay of that conserved C and broken P implies that CP is broken. This is the to achieve an angular asymmetry. The driving term is an the breaking pattern caused by the QCD theta-vacuum-angle term. EDTM. For our concrete estimates we have used the CP violating What about the case when CP is conserved? This is fulfilled when =−¯ QCD theta-vacuum-angle term. Since this term gives also rise to bW bW (i.e. bW is purely real) [9]. For a P violating decay, an EDM of the neutron one can use chiral perturbation theory to conservation of CP implies that Cis violated. Note that for the case relate the two EDMs. Using the experimental upper limit for the when P, C and CP are violated bW is neither purely real nor purely neutron EDM provides upper limits for the angular asymmetry in imaginary. our 0 decay and also for the parameter that tests CP violation by Finally we recall that electromagnetic final-state interactions in- comparing the angular asymmetries for particle and antiparticle. duce a phase shift δF that is the same for particle and antiparticle. We have found tiny effects which implies that any experimental Thus significance would point to physics beyond QCD, even if extended = iδF ¯ = ¯ iδF by a theta-vacuum-angle term. b bW e , b bW e . (A.2) S.S. Nair et al. / Physics Letters B 788 (2019) 535–541 541

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